The Hopf-Tsuji-Sullivan theorem states that the geodesic flow on
(an infinite) Riemann surface is ergodic iff the Poincare series is
divergent iff the Brownian motion is recurrent. Infinite Riemann
surfaces can be built by gluing infinitely many...
Let Ω be an open set in a Euclidean space X of dimension (n+1)
and ϕ be a uniformly convex smooth norm on X. Consider an
n-dimensional unit-density varifold V in Ω, whose generalised mean
curvature vector, computed with respect to ϕ, is bounded...
The mathematical core of deep learning is function approximation
by neural networks trained on data using stochastic gradient
descent. I will present a collection of sharp results on training
dynamics for the deep linear network (DLN), a...
Several recent groundbreaking results in geometric measure
theory, homogeneous dynamics and number theory ultimately rely on a
key result of Bourgain known as Bourgain's Projection Theorem (of
course, each of these results require many other tools...
We use the min-max construction to find closed hypersurfaces
which are stationary with respect to anisotropic elliptic
integrands in any closed n-dimensional manifold . These surfaces
are regular outside a closed set of zero n-3 dimension. The...
In the early 80s Hatcher proved the Smale Conjecture, asserting
that the diffeomorphism group of the three-sphere retracts onto its
isometry group. The corresponding problem for RP^3 was open
nearly 40 years, and resolved only in 2019 by a detailed...
I will present recent work with Hairer, Rosati and Yi
establishing quantitative lower bounds for the top Lyapunov
exponent of linear PDEs driven by two-dimensional stochastic
Navier-Stokes equations on the torus. For both the
advection-diffusion...
Parallels between elliptic and parabolic theory of partial
differential equations have long been explored. In particular,
since elliptic theory can be seen as a steady-state version of
parabolic theory, if a parabolic estimate holds, then by...
In this talk, I will discuss some results concerning the
geometry and topology of manifolds on which the first eigenvalue of
the operator -γΔ + Ric is bounded below. Here, γ is a positive
number, Δ is the Laplacian, and Ric denotes the pointwise...
